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A321227
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Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.
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2
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0, 1, 3, 6, 17, 43, 147, 458, 1729, 6445, 27011
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OFFSET
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0,3
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COMMENTS
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The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(4) = 17 multiset partitions:
{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1,2}} {{1,1,2}} {{1,1,1,2}}
{{1},{1}} {{1,2,3}} {{1,1,2,2}}
{{1},{1,1}} {{1,1,2,3}}
{{1},{1,2}} {{1,2,3,4}}
{{1},{1},{1}} {{1},{1,1,1}}
{{1,1},{1,1}}
{{1},{1,1,2}}
{{1,1},{1,2}}
{{1},{1,2,2}}
{{1},{1,2,3}}
{{1,2},{1,3}}
{{2},{1,1,2}}
{{1},{1},{1,1}}
{{1},{1},{1,2}}
{{1},{2},{1,2}}
{{1},{1},{1},{1}}
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
mensity[c_]:=Total[(Length[Union[#]]-1&)/@c]-Length[Union@@c];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Sum[Length[Select[mps[m], And[mensity[#]==-1, Length[csm[#]]==1]&]], {m, strnorm[n]}], {n, 0, 8}]
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CROSSREFS
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Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321228, A321229, A321231.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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